Official Math 112 Catalog Description

Transcription

1 Official Math 112 Catalog Description Topics include properties of functions and graphs, linear and quadratic equations, polynomial functions, exponential and logarithmic functions with applications. A graphing calculator is required for this course. We recommend the TI-83 or TI-84 models. Calculators that perform symbolic manipulations, such as the TI-89, NSpire CAS, or HP50g, cannot be used. Except as per University policy on repeating a course, credit will not be given for this course if the student has credit in a higher level math course. Such students may be dropped from the course. Examinations are proctored. Section 1.1 Math 112 Course Objectives Determine if a given equation is linear. Solve linear equations of all types. Solve equations that may be rewritten to be linear. Section 1.2 Write mathematical statements given a verbal description. Set up a linear equation from a verbal description and solve. Set up and solve mixing problems. Set up and solve problems involving distance, rate, and time. Set up and solve shared work problems. Section 1.4 Use the Zero Product Property. Solve a quadratic equation by factoring. Solve a quadratic equation by completing the square. Solve a quadratic equation by using the quadratic formula. Use the sign of the discriminant to determine the number of solutions for a quadratic equation.

2 Section 1.5 Use the four-step problem solving strategy to solve application problems involving quadratic equations. Solve quadratic application problems involving unknown numerical quantities. Solve quadratic application problems involving geometric shapes. Solve quadratic application problems involving projectile motion. Solve quadratic application problems involving distance, rate, and time. Solve quadratic application problems involving two people working together. Section 2.3 Determine the slope of a line, given two points on the line. Sketch the graph of a line, given a point and the slope, or given the equation of the line. Find the equation of a line, given two points, or given one point and the slope. Find the slope and/or y-intercept of a line, given the equation of the line. Find the equation of a horizontal or vertical line passing through a given point. Section 2.4 Determine whether two lines given by equations are parallel, perpendicular, or neither. Given the equation of a line, determine the equation of a parallel line passing through a given point. Given the equation of a line, determine the equation of a perpendicular line passing through a given point.

3 Section 3.1 Understand the definitions of relation and function. Determine whether a relation is a function. Given a table of values, an equation in two variables, or a graph, determine whether it represents a function or a relation, and explain your reasoning. Identify the domain of a function. Use function notation correctly. Evaluate a function at a value in its domain. Evaluate a function at an algebraic expression. Calculate a difference quotient. Use the vertical line test to determine whether a graph represents a function. Understand the general form for polynomial functions. Understand the general form for rational functions. Understand the general form for root functions. Determine the domain of a function given an equation. Understand and properly use the phrase is a function of. Section 3.2 Find the intercepts of a function given its formula. Given the graph of a function, identify the domain and range of the function. Approximate relative minimum and maximum values and intercepts from a graph. Given the graph of a function, identify open intervals where the graph is increasing/decreasing/constant. Given the graph of a function, identify intervals where the graph is positive/negative. Given the equation or graph of a function, determine whether the function is odd, even, or neither. Use a calculator to approximate features on a graph, such as zeros, maximums, and minimums. Use a calculator to answer questions about a function in a given context. Given certain characteristics of a function, such as the domain, range, and/or intervals on which it is increasing/decreasing or positive/negative, sketch a possible graph of the function. Given a partial table of values for an even/odd function, complete the table. Given a partial graph of an even/odd function, complete the graph. Sketch a complete graph of a function which includes all important features. Using your knowledge of domain and range, as well as context provided, find an appropriate viewing window for a function on a graphing calculator.

4 Section 3.3 Identify by sight the graphs of basic functions, such as the constant function, the identity function, the square function, the cube function, the absolute value function, the square root function, the cube root function, the reciprocal function, and the greatest integer function. Evaluate a piecewise-defined function at a value in its domain. Graph a piecewise-defined function. Given the graph of a piecewise-defined function, find a rule that describes the function. Solve application problems involving piecewise-defined functions. Section 3.4 Given a function represented by an equation, write an equation that translates the graph of the function up/down, left/right. Given a function represented by an equation, write an equation that reflects the graph of the function across the x axis and/or y axis. Given a function represented by an equation, write an equation that expands or compresses the graph of the function horizontally and/or vertically. Given an equation that represents a transformation of a basic function, identify the transformation(s) involved, and graph the new function. Given a function represented by a table of values, and given a series of transformations to that function, identify the transformations involved, and determine the table of values for the new function. Given the graph of a transformation of a basic function, find the equation for that function by analyzing the transformation(s) performed, and the order of the transformations. Section 3.5 Find the sum, difference, product, or quotient of functions represented graphically, algebraically, or in table form. Find the domain of the sum, difference, product, or quotient of functions. Compose two functions represented graphically, algebraically, or in table form. Given a compound function, identify two functions whose composition yields the given function. Find the domain of a composition of two functions.

5 Section 3.6 Given a function represented by a graph, an equation, a table of values, or a verbal description, determine whether the function is one to one. Verify algebraically that two functions are inverse functions by finding the composition of the two functions. Given the graph of a one to one function, identify the domain and range of the inverse function, and sketch its graph. Given the domain and range of a one to one function, or given a one to one function for which you can identify the domain and range, determine the domain and range of the inverse function. Find the inverse function of a one to one function represented by a graph, an equation, or a table of values. Section 4.1 Given a function represented algebraically, determine whether the function is quadratic. Convert a quadratic function from general form to standard form. Identify and understand the components of a quadratic function in standard form, and identify the impact of these components on the graph of the function. Complete the square for a quadratic function in general form. Identify the vertex or maximum/minimum of a quadratic function using either the standard form of the function or the vertex formula. Given the graph of a quadratic function, find the equation of the function. Find the equation of a quadratic function, given the vertex and one other point. Find a possible equation of a quadratic function, given two zeros. Section 4.2 Set up a quadratic function to model certain "real life" situations. Use quadratic models and interpret results in "real life" terms. Understand projectile motion models. Understand area models. Understand demand, cost, revenue and profit examples from business.

6 Section 4.3 Understand the general form for a polynomial function. Identify polynomial functions. Sketch graphs of power functions. Use transformations to sketch the graph of certain polynomial functions. Determine the end behavior of polynomials from the leading term property. Use the end behavior of a polynomial to determine the possible degree of the polynomial and sign of the leading coefficient. Determine the intercepts of the graph of a polynomial function. Determine the zeros of a polynomial function and their multiplicities. Sketch the graph of a polynomial function. Determine a possible equation of a polynomial function given its graph. Section 4.4 Use the division algorithm to divide two polynomials by long division. Use synthetic division to divide a polynomial by a linear factor of the form x c. Use the Remainder Theorem. Use the Factor Theorem. Factor polynomials and sketch their complete graph.

7 Section 4.6 Identify whether a function represented by an equation is a rational function. Determine the domain of a rational function represented by an equation or a graph. Given a rational function in the form f(x), use polynomial division to rewrite in the form g(x) q(x) + r(x), where q(x) is the quotient, and r(x) is the remainder when f(x) is divided d(x) by g(x). Given a rational function represented by an equation, determine the equation of any vertical asymptote for the function. Given a rational function represented by an equation, determine whether the function has a horizontal asymptote, a slant asymptote, or neither. Given a rational function represented by an equation, find the equation of the horizontal asymptote, if it exists. Given a rational function represented by an equation, find the equation of the slant asymptote, if it exists, by using polynomial long division. Use the horizontal or slant asymptote of a rational function to determine the end behavior of the function. Given a rational function represented by an equation, determine whether the graph of a rational function has a hole, and find the x-value at which the hole in the graph occurs. Given the graph of a rational function, identify and use all important features of the graph to find a possible equation for the function. Section 5.1 Identify exponential functions. Understand the characteristics of exponential functions of the form f(x) = b x, including the domain, range, intercept, asymptote, end behavior, and general graphs. Determine the formula for an exponential function given its graph. Sketch the graph of exponential functions using transformations. Solve exponential equations by relating the bases. Solve compound interest application problems. Determine the present value of an investment. Solve exponential application problems

8 Section 5.2 Understand the characteristics of the natural exponential function f(x) = e x, including the domain, range, intercept, asymptote, end behavior, and its graph. Sketch the graph of natural exponential functions using transformations. Solve natural exponential equations by relating the bases. Solve continuous compound interest application problems. Determine the present value of an investment using continuous compound interest. Solve population growth application problems. Section 5.3 Apply the definition of a logarithm to convert an equation in logarithmic form into exponential form. Given an equation in exponential form, rewrite the equation in logarithmic form. Properly identify the notation for natural (base e) and common (base 10) logarithm. Evaluate natural and common logarithms on a calculator. Know and identify the shape and basic features of the graph of f(x) = ln x and f(x) = ln x. Evaluate simple logarithms of any allowable base without the use of a calculator. Determine the domain of logarithmic functions. Section 5.4 Apply the properties of logarithms to expand expressions involving the logarithm of a product/quotient/power into a sum/difference of logarithms. Apply the properties of logarithms to condense a sun/difference of logarithms into a single logarithm. Use the change of base formula to convert a logarithm with any allowable base to a logarithm with any allowable base. (In particular, convert to base 10 or base e.) Use the one-to-one property of logarithms to solve certain logarithmic equations. Section 5.5 Solve exponential equations of all types. Solve logarithmic equations of all types.

9 Section 5.6 Solving exponential and logarithmic equations in the context of applications, including compound interest, doubling time, half-life, logistic growth model, Newton's Law of Cooling, and others. Set up and solve application problems involving exponential growth and decay.